This follows from a simple extension of the so-called cyclic lemma (see e.g. [1]}). Indeed, if \({x}=(a_{i},b_{i})_{1 \leqslant i \leqslant m}\)  are integers such that \(a_{1}+a_{2}+\cdots +a_{m}=-1\) , then there is a unique \(j \in \mathbb {Z}/m\mathbb {Z}\)  such that the cyclic shift \({x}^{(j)}=(a^{(j)}_{i},b^{(j)}_{i})_{1 \leqslant i \leqslant m}\)  fulfills \(a^{(j)}_{1}+ \cdots +a^{(j)}_{i} \geqslant 0\)  for every \(1 \leqslant i<m\) . It is then standard to obtain the desired results by an exchangeability argument; we leave details to the reader.
